A common misconception that seems to pop up all the time. No, a Kindle device itself doesn't automatically grant you a library of free books. The confusion usually stems from bundling 'Amazon Prime' with 'Kindle Unlimited'—they're completely separate subscriptions with their own costs.
Owning a Prime membership does give you access to a specific, rotating selection called Prime Reading. It's not the entire Kindle store for free; it's more like a curated shelf of maybe a thousand titles, including some popular reads, first books in series, and a lot of self-published works. I find myself checking it every month or so, but it's rarely where I find my most anticipated reads.
For a vast library, you'd need Kindle Unlimited, which is a separate monthly fee. The real value for Prime members, honestly, is often in the 'First Reads' program where you can get one pre-release book for free each month from a small editors' choice list. That's a decent perk, but it's a far cry from an all-you-can-read buffet.